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# Optimization Methods in Computational Fluid Dynamics

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Optimization Methods in Computational Fluid Dynamics
Antony Jameson and, assisted by, Kui Ou
Aeronautics and Astronautics Department, Stanford University, Stanford, CA, USA 1 Introduction: Aerodynamic Design 1 2 Aerodynamic Optimization and Control Theory 2 3 Formulation of the Design Problem as a Control Problem 3 4 Design Optimization Procedure 5 5 Airfoil Design for Potential Flow using Conformal Mapping 5 6 Wing Design using the Euler Equations 7 7 Design Case Studies 9 8 Conclusion 12 Acknowledgment 13 References 13
1 INTRODUCTION: AERODYNAMIC DESIGN
In order to set the context for aerodynamic shape opti- mization, the airplane design process is summarized in the following paragraphs. It can generally be divided into three phases: conceptual design, preliminary design, and final detailed design, as illustrated in Figure 1. The conceptual design stage defines the mission in the light of anticipated market requirements and determines a general preliminary configuration capable of performing this mission, together
Encyclopedia of Aerospace Engineering. Edited by Richard Blockley and Wei Shyy c�� 2010 John Wiley & Sons, Ltd. ISBN: 978-0-470-68665-2
with first estimates of size, weight, and performance. In the preliminary design stage, the aerodynamic shape and structural skeleton progress to the point where detailed per- formance estimates can be made and guaranteed to potential customers. At this stage, the development costs are still fairly moderate, in the range of 50–100 million dollars. In the final design stage, the structure must be defined in complete detail, together with complete systems, including the flight deck,
Conceptual design Preliminary design Final design Defines mission preliminary sizing weight, performance
Figure 1. The overall design process.

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2 Computational Fluid Dynamics
{Propulsion
Noise Stability Control Loads Structures Fabrication Conceptual design CAD definition Mesh generation CFD analysis Visualization Performance evaluation Multi−disciplinary evaluation Wind tunnel testing Central database Detailed final design Release to manufacturing Model fabrication
Outer loop Major design c y c le Inner loop
Figure 2. The aerodynamic design process.
control systems (involving major software development for fly-by-wire systems), electrical and hydraulic systems, land- ing gear, weapon systems for military aircraft, and cabin layout for commercial aircraft. Major costs are incurred at this stage, during which it is also necessary to prepare a detailed manufacturing plan, together with appropriate facilities and tooling. The development costs to reach the point of initial production are in the range of 3–10 billion dollars. Thus, the final design would normally be carried out only if suffi- cient orders have been received to indicate a reasonably high probability of recovering the investment. In the development of commercial aircraft, aerodynamic design plays a leading role in the preliminary design stage. The definition of the external aerodynamic shape may actu- ally be finalized in the preliminary design. The aerodynamic lines of the Boeing 777 were frozen, for example, when ini- tial orders were accepted before the initiation of the detailed design of the structure. Figure 2 illustrates the way in which the aerodynamic design process is embedded in the overall preliminary design. The starting point is an initial Com- puter Aided Design (CAD) definition resulting from the conceptual design. The inner loop of aerodynamic analysis is contained in an outer multi-disciplinary loop, which is in turn contained in a major design cycle involving wind tunnel testing. In recent Boeing practice, three major design cycles, each requiring about 4–6 months, have been used to finalize the wing design.
2 AERODYNAMIC OPTIMIZATION AND CONTROL THEORY
The use of computational simulation to scan many alterna- tive designs has proved extremely valuable in practice, but it is also evident that the number of possible design variations is too large to permit their exhaustive evaluation, and thus it is very unlikely that a truly optimum solution can be found without the assistance of automatic optimization procedures. To ensure the realization of the true best design, the ulti- mate goal of computational simulation methods should not just be the analysis of prescribed shapes but the automatic determination of the true optimum shape for the intended application. The need to find optimum aerodynamic designs was already well recognized by the pioneers of classical aerody- namic theory. A notable example is the determination that the optimum span-load distribution that minimizes the induced drag of a monoplane wing is elliptic (Glauert, 1926; Prandtl and Tietjens, 1934). There are also a number of famous results for linearized supersonic flow. The body of revolution of min- imum drag was determined by Sears (1947), while conditions for minimum drag of thin wings due to thickness and sweep were derived by Jones (1981). The problem of designing a two-dimensional profile to attain a desired pressure distribu- tion was studied by Lighthill (1945), who solved it for the case of incompressible flow with a conformal mapping of the profile to a unit circle. The speed over the profile is q = 1 h |∇��| (1) where �� is the potential that is known for incompressible flow and h is the modulus of the mapping function. The surface value of h can be obtained by setting q = qd, where qd is the desired speed, and since the mapping function is analytic, it is uniquely determined by the value of h on the boundary. A solution exists for a given speed q�� at infinity only if 1 2��
��
q d�� = q�� (2) and there are additional constraints on q if the profile is required to be closed. These classical results were limited to special cases amenable to analytic methods and could not readily be

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3 FORMULATION OF THE DESIGN PROBLEM AS A CONTROL PROBLEM
The simplest approach to optimization is to define the geom- etry through a set of design parameters, which may, for example, be the weights ��i applied to a set of shape functions bi(x) so that the shape is represented as f(x) =
��
��ibi(x) (3) Then, a cost function I is selected, which might, for exam- ple, be the drag coefficient or the lift-to-drag ratio, and I is regarded as a function of the parameters ��i. The sensitivities ∂I/∂��i may now be estimated by making a small variation�Ħ�i in each design parameter in turn and recalculating the flow

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[
∂IT w
]
��w +
[
∂IT F
]
��F (9) in the cost function. Suppose that the governing equation R that expresses the dependence of w and F within the flow- field domain D can be written as R (w,F) = 0 (10) Then, ��w is determined from the equation ��R =
[
R w
]
��w +
[
R F
]
��F = 0 (11) Since the variation ��R is zero, it can be multiplied by a Lagrange multiplier �� and subtracted from the variation ��I without changing the result. Thus, equation (9) can be replaced by ��I = ∂IT w ��w + ∂IT F ��F ��T
([
R w
]
��w +
[
R F
]
��F
)
=
{
∂IT w��T
[
R w
]}
��w +
{
∂IT F��T
[
R F
]}
��F (12) Choosing �� to satisfy the adjoint equation
[
R w
]T
�� = ∂I w (13) the first term is eliminated, and we find that ��I = GT��F (14) where G = ∂IT F��T
[
R F
]
(15) The advantage is that (14) is independent of ��w, with the result that the gradient of I with respect to an arbitrary num- ber of design variables can be determined without the need for additional flow-field evaluations. In the case that (10) is a partial differential equation, the adjoint equation (13) is also a partial differential equation and determination of the appropriate boundary conditions requires careful mathemati- cal treatment. In the case of the discrete adjoint method, these formulas are applied directly to the discretized cost function and flow equations.

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Optimization Methods in Computational Fluid Dynamics 5
4 DESIGN OPTIMIZATION PROCEDURE
Once the gradient G has been determined, an improvement can be obtained by making a shape change ��F = −��G (16) where �� is sufficiently small and positive so that ��I = −��GTG < 0 (17) In the infinitely dimensional case where the shape is repre- sented by a functionF(x), the gradient is also a functionG(x), and the increment of the cost is represented by the integral ��I =
��
G(x)��F(x)dx (18) Now, it becomes important to maintain the smoothness of the shape. For this purpose, one can set ��F = −��¯G (19) where ¯G satisfies the smoothing equation ¯G ∂x ϵ ∂x ¯G = G, ¯G = 0 at end points (20) Then ��I =
��
G(x)��F(x)dx = −��
��
G ∂x ϵ ∂x ¯G) ¯G dx (21) or, integrating by parts ��I = −��
��(
¯G2 + ϵ
(
∂x ¯G
)2 )
dx (22) so an improvement is still guaranteed. The smoothed gradient ¯G can be regarded as the gradient with respect to a modified Sobolev inner product (u, v) =
��
(uv + ϵu v )dx (23) When second-order central differencing is applied to (20), the equation at a given node, i, can be expressed as ¯Gi ϵ
(
¯Gi+1 − 2¯Gi + ¯Gi−1
)
= Gi, 1 �� i �� n (24) where Gi and ¯Gi are the point gradients at node i before and after the smoothing, respectively, and n is the number of design variables equal to the number of mesh points in this case. Then, ¯G = AG (25) where A is the n �� n tridiagonal matrix such that A
−1 =
     
1 + 2ϵ ϵ 0 . 0 ϵ . . 0 . . . . . .ϵ 0 ϵ 1 + 2ϵ
     
(26) Then, in each design iteration, a step, ��F, is taken such that ��F = −��AG (27) As can be seen from the form of this expression, implicit smoothing may be regarded as a preconditioner that allows the use of much larger steps for the search procedure and leads to a large reduction in the number of design iterations needed for convergence. The design procedure can finally be summarized as follows: 1. Solve the flow equations for w. 2. Solve the adjoint equations for �� subject to appropriate boundary conditions. 3. Evaluate G and calculate the corresponding Sobolev gradient ¯G. 4. Project ¯G into an allowable subspace that satisfies any geometric constraints. 5. Update the shape based on the direction of steepest descent. 6. Return to 1 until convergence is reached. The application of this method is illustrated for the cases of transonic potential flow and flow described by the Euler equations in the next two sections.
5 AIRFOIL DESIGN FOR POTENTIAL FLOW USING CONFORMAL MAPPING
Consider the case of two-dimensional compressible inviscid flow. In the absence of shock waves, an initially irrotational flow will remain irrotational, and we can assume that the velocity vector q is the gradient of a potential ��. In the

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6 Computational Fluid Dynamics presence of weak shock waves this remains a fairly good approximation. Let p, ��, c, and M be the pressure, density, speed of sound, and Mach number q/c. Then, the potential flow equation is ∇ · (����) = 0 (28) where the density is given by �� =
[
1 + �� − 1 2 M2
��
(
1 − q2)
] 1
��−1
(29) while p = ���� ��M2
��
, c2 = ��p �� (30) Here,M�� is the Mach number in the freestream, and the units have been chosen so that p and q have a value of unity in the far field. Suppose that the domain D exterior to the airfoil profile C in the z plane is conformally mapped on to the domain exterior to a unit circle in the �� plane as sketched in Figure 3. Let R and �� be polar coordinates in the �� plane, and let r be the inverted radial coordinate 1/R. Also, let h be the modulus of the derivative of the mapping function h =
∣ ∣ ∣ ∣
dz d��
∣ ∣ ∣ ∣
(31) Now the potential flow equation becomes ∂�� (�Ѧ���) + r ∂r (r�Ѧ�r) = 0 in D (32) where the density is given by equation (29), and the circum- ferential and radial velocity components are u = r���� h , v = r2��r h (33)
(a) C D (b)
Figure 3. Conformal mapping. (a) z plane; (b) �� plane.
while q2 = u2 + v2 (34) The condition of flow tangency leads to the Neumann bound- ary condition v = 1 h ∂�� ∂r = 0 on C (35) In the far field, the potential is given by an asymptotic esti- mate, leading to a Dirichlet boundary condition at r = 0. Suppose that it is desired to achieve a specified velocity distribution qd on C. Introduce the cost function I = 1 2
��
C
(q qd)2 d�� (36) The design problem is now treated as a control problem where the control function is the mapping modulus h, which is to be chosen to minimize I subject to the constraints defined by the flow equations (28–35). A modification ��h to the mapping modulus will result in variations �Ħ�, ��u, ��v, and �Ħ� to the potential, velocity com- ponents, and density. The resulting variation in the cost will be ��I =
��
C
(q qd) ��q d�� (37) where, on C, q = u. Also, ��u = r �Ħ��� hu ��h h , ��v = r2 �Ħ�r hv ��h h (38) while according to equation (29) ∂�� ∂u = − ��u c2 , ∂�� ∂v = − ��v c2 (39) It follows that �Ħ� satisfies L�Ħ� = − ∂��
(
��M2���� ��h h
)
r ∂r
(
��M2r��r ��h h
)
(40) where L �� ∂��
[
��
(
1 − u2 c2
)
∂����uv c2 r ∂r
]
+ r ∂r
[
��
(
1 − v2 c2
)
r ∂r��uv c2 ∂��
]
(41)

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Optimization Methods in Computational Fluid Dynamics 7 Then, if�� is any periodic differentiable function that vanishes in the far field,
��
D
�� r2 L �Ħ� dS =
��
D
��M2∇�� · ∇�� ��h h dS (42) where dS is the area element r dr d��, and the right-hand side has been integrated by parts. Now, we can augment equation (37) by subtracting the constraint (42). The auxiliary function �� then plays the role of a Lagrange multiplier. Thus, ��I =
��
C
(q qd)q ��h h d��
��
C
�Ħ� ∂��
(
q qd h
)
d��
��
D
�� r2 L�Ħ� dS +
��
D
��M2∇�� · ∇�� ��h h dS (43) Now suppose that �� satisfies the adjoint equation L�� = 0 in D (44) with the boundary condition ∂�� ∂r = 1 �� ∂��
(
q qd h
)
on C (45) Then, integrating by parts, ��I = −
��
C
(q qd)q ��h h d�� +
��
D
��M2∇�� · ∇�� ��h h dS (46) Here, the first term represents the direct effect of the change in the metric, while the area integral represents a correc- tion for the effect of compressibility. When the second term is deleted, the method reduces to a variation of Lighthill��s method (Lighthill, 1945). Equation (46) can be further simplified to represent ��I purely as a boundary integral because the mapping function is fully determined by the value of its modulus on the boundary. Set log dz d�� = F + i�� (47) where F = log
∣ ∣ ∣ ∣
dz d��
∣ ∣ ∣ ∣
= logh (48) and ��F = ��h h (49) Then, F satisfies Laplace��s equation F = 0 in D (50) and if there is no stretching in the far field, F �� 0. Introduce another auxiliary function P which satisfies P = ��M2∇�� · ∇�� in D (51) and P = 0 on C (52) Then, after integrating by parts, we find that ��I =
��
C
G��Fc d�� (53) where Fc is the boundary value of F, and G = ∂P ∂r − (q qd)q (54) Thus, we can attain an improvement by a modification ��Fc = −��¯G (55) in the modulus of the mapping function on the boundary, which in turn defines the computed mapping function sinceF satisfies Laplace��s equation. In this way, the Lighthill method is extended to transonic flow. More details can be found in the studies by Jameson (1988) and Jameson and Martinelli (1999).
6 WING DESIGN USING THE EULER EQUATIONS
The application of control theory to aerodynamic design problems is illustrated in this section for the case of three- dimensional wing design using the compressible Euler equations as the mathematical model. The extension of the method to treat the Navier–Stokes equations is presented in references Jameson, Martinelli and Pierce (1998), Jameson and Martinelli (1999), and Jameson (2003). It proves con- venient to denote the Cartesian coordinates and velocity

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8 Computational Fluid Dynamics components by x1, x2, x3 and u1, u2, u3 and to use the conven- tion that summation over i = 1 to 3 is implied by a repeated index i. Then, the three-dimensional Euler equations may be written as w ∂t + fi ∂xi = 0 (56) where w =
       
�� ��u1 ��u2 ��u3 ��E
       
, fi =
       
��ui ��uiu1 + p��i1 ��uiu2 + p��i2 ��uiu3 + p��i3 ��uiH
       
(57) and ��ij is the Kronecker delta function. Also p = (�� − 1)��
(
E − 1 2 u2
i
)
(58) and ��H = ��E + p (59) where �� is the ratio of the specific heats. In order to simplify the derivation of the adjoint equations, we map the solution to a fixed computational domain D with coordinates ��1, ��2, ��3 where Kij =
[
∂xi ∂��j
]
, J = det (K), K
−1 ij
=
[
∂��i ∂xj
]
(60) and S = JK
−1
(61) The elements of S are the cofactors of K, and in a finite volume discretization they are just the face areas of the com- putational cells projected in the x1, x2, and x3 directions, while J represents the cell volume. Now, multiplying equation(56) by J and applying the chain rule, J w ∂t + R (w) = 0 (62) where R (w) = Sij fj ∂��i = ∂��i
(
Sijfj
)
(63) We can write the transformed fluxes in terms of the scaled contravariant velocity components Ui = Sijuj (64) as Fi = Sijfj =
       
��Ui ��Uiu1 + Si1p ��Uiu2 + Si2p ��Uiu3 + Si3p ��UiH
       
(65) For convenience, the coordinates ��i describing the fixed computational domain are chosen so that each boundary conforms to a constant value of one of these coordinates. Variations in the shape then result in corresponding variations in the mapping derivatives defined by Sij. Moreover, it will be assumed that the portion of the boundary that undergoes shape modifications is restricted to the coordinate surface ��2 = 0. In order to illustrate the procedure, consider an inverse problem where we aim to attain a desired surface pressure pd. Then, a natural choice is to define the cost function as I = 1 2
��
B
(p pd)2 dS (66) where pd is the desired surface pressure, and the integral is evaluated over the actual surface area of the wing or any other shape that is to be redesigned. In the computational domain this is transformed to I = 1 2
�� ��
B
(p pd)2 |S2|d��1d��3 (67) where the quantity |S2| =
��
S2jS2j (68) denotes the face area corresponding to a unit element of face area in the computational domain. In the steady state, it follows from equation (63) that the variation ��w in the solution due to a shape change is deter- mined by ��R = ∂��i ��Fi = 0 (69) where ��Fi = Sij Fj w ��w + ��Sijfj (70)

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Optimization Methods in Computational Fluid Dynamics 9 Here, the shape change appears in the metric variations ��Sij. Now we multiply this by a Lagrange multiplier ��T, generally called a costate vector, and subtract its integral over the flow domain from the variation in the cost to obtain ��I =
�ҡ�
B
(p pd)��p|S|d��1��3 + 1 2
�ҡ�
B
(p pd)2 |��S|2 d��1d��3 −
��
D
��T ∂��i ��Fid�� (71) Next, the last term may be integrated by parts to give −
�� ��
B
��T��F2d��1d��3 −
��
��T ∂�� (Sij fj w )��wd��
��
D
��T ∂�� ��Sijfj(w)d�� (72) where contributions from the far-field boundaries have been eliminated by appropriate choice of the far-field adjoint boundary conditions (Jameson (1988)). Now it can be seen that any explicit dependence of the cost variation ��I on ��w can be eliminated by requiring the costate vector �� to satisfy the adjoint equation
[
Sij f w
]T ��
∂��i = 0 in D (73) and also canceling the first boundary integral containing ��p by the boundary integral containing ��F2. Now the condition that there is no flow through the wall boundary at ��2 = 0 is equivalent to U2 = 0 (74) so that ��U2 = 0 (75) when the boundary shape is modified. Consequently, the variation of the inviscid flux at the boundary reduces to ��F2 = ��p
            
0 S21 S22 S23 0
            
+ p
            
0 ��S21 ��S22 ��S23 0
            
(76) Accordingly, the desired cancellation is achieved by requiring �� to satisfy the boundary condition ��jnj = p pd (77) where nj are the components of the surface normal. The variation in the cost is thus reduced to ��I = 1 2
�ҡ�
B
(p pd)2 |��S|2 d��1d��3 −
�ҡ�
B
(��S21��2 + ��S22��3 +��S23��4) p d��1d��3 +
��
D
��T ∂�� ��Sijfj(w)d�� (78) Finally, an explicit formulation for the gradient can be extracted by defining the way in which the mesh is deformed when the shape is modified and hence the metric perturbations ��Sij. Other cost functions can be addressed by appropriate modification of the adjoint boundary condition to cancel the boundary integral in the cost variation. Since ��F2 depends only on the pressure variation ��p and the metric variations, any cost function involving an integral of the surface pressure can be treated. Some representative examples are presented in the following section.
7 DESIGN CASE STUDIES
7.1 Two-dimensional studies of transonic airfoil design
When the inviscid Euler equations are used to model the flow, the source of drag is the wave drag due to shock waves. Accordingly, if the shape is optimized for minimum drag at fixed lift, the best attainable result is a shock-free airfoil with zero drag. By this criterion, the optimum shape is com- pletely non-unique, since all shock-free profiles are equally good. Experience during the last 15 years has confirmed that shock-free profiles can be obtained from a wide variety of initial shape, while maintaining a fixed lift coefficient and a fixed thickness. An example of shock-free design is shown in Figure 4 for the optimization of a DLBA-243 airfoil (Harbeck and Jameson, 2005).
7.2 Shark Reno Racer design
Using three-dimensional optimization, it is possible to arrive very rapidly at an efficient design. This is illustrated in the next example that is the wing design for a proposed

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10 Computational Fluid Dynamics
Figure 4. Pressure distribution and Mach contours for the DLBA-243 airfoil. (a) Before optimization; (b) after optimization.
propeller-driven Reno Air Racer, the Shark, which was intended to reach speeds of 550 mph (Jameson and Vassberg, 2001). A sketch of Shark is shown in Figure 5a. The initial wing had extremely strong shock waves that were removed by optimization using the Euler equations, as shown in Figure 5b and 5c. Then the design was fur- ther refined by inverse design with the Reynolds-averaged Navier–Stokes equations to produce a pressure distribution that could promote laminar flow over a range of lift coeffi- cients as shown in Figure 5d.

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Optimization Methods in Computational Fluid Dynamics 11
Figure 5 . Reno Racer W ing design optimization using Euler equations and inv erse design with Navier–Stok es equations.(a) Side view o f Shark design;(b) shock elimination with optimization; (c) optimization results with Euler equations; and (d) results of Navier–Stok es inv erse design.

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12 Computational Fluid Dynamics
(a) (b)
0.0E+00 0.1E+00 0.2E+00 0.3E+00 0.4E+00
CL
0.5E+00 0.6E+00 0.0E+00 0.8E+01 0.2E+02 0.2E+02 0.3E+02
L /D CD (counts) Mach Drag rise at fixed CL CD (counts)
0.4E+02 0.5E+02 0.00 50.00 100.00 150.00 200.00 250.00 300.00
280 260 240 220 200 180 160 140 120 100 0.78 0.8 0.82 0.84 0.86 0.88 0.9 0.92
Baseline Redesign
Figure 6. (a) Drag Polars of Baseline and Super B747 at Mach 0.86; (b) Drag vs. Mach no. of Super B747. Solid line represents Super B747. Dash line represents Baseline B747.
7.3 Super B747
In order to obtain the maximum possible benefit, one should allow for redesign of the planform as well as the wing sec- tion shapes. Then, it is necessary to estimate the change in the wing weight that will result from the redesign. The next example, a new wing design for a ��Super Boeing 747��, with an increased span, reduced sweep, and a much thicker wing section to reduce the weight, illustrates the kind of improve- ments that can be obtained by optimization. The numbers in Figure 6a show, while the curve in Figure 6b compares the drag rise of the ��Super B747�� with the existing B747 (Leoviriyakit and Jameson, 2004).
7.4 Shape optimization of a transonic business jet
The same design method has also been applied to several complete aircraft configurations using unstructured meshes (Jameson, Shankaran and Martinelli, 2008). The results for a business jet are shown in Figure 7a and 7b. There is a
Figure 7. Density contours for a business jet at M = 0.8, �� = 2◦ . (a) Baseline; (b) Redesign.
strong shock over the outboard wing sections of the initial configuration, which is essentially eliminated by the redesign. The drag was reduced from 235 counts to 215 counts in about eight design cycles. The lift was constrained at 0.4 by per- turbing the angle of attack. Further, the original thickness of the wing was maintained during the design process ensuring that fuel volume and structural integrity will be maintained by the redesigned shape. Thickness constraints on the wing were imposed on cutting planes along the span of the wing and by transferring the constrained shape movement back to the nodes of the surface triangulation.
8 CONCLUSION
The adjoint design method presented in these notes is now well established and has proved effective in a variety of indus- trial applications including, most recently, the wing design of the Gulfstream G650. The method combines the versatility of numerical optimization methods with the efficiency of inverse design methods. The geometry is modified by a

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Optimization Methods in Computational Fluid Dynamics 13 grid perturbation technique, which is applicable to arbitrary configurations. The combination of computational efficiency with geometric flexibility provides a powerful tool, with the final goal being to create practical aerodynamic shape design methods for complete aircraft configurations. Motivated by the need to optimize complete configurations, the further development of adjoint methods on unstructured meshes is a subject of ongoing research (Nielsen and Park, 2006; Castro and Zuazua, 2007; Mavriplis, 2007). It may eventually prove most effective to incorporate CAD definitions of the geometry directly into the shape modification procedure.
ACKNOWLEDGMENT
The author��s research on shape optimization has benefited greatly from the continuing support over more than 20 years of the Air Force Office of Scientific Research through its Computational Mathematics Program.
REFERENCES
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